Hmm, allow me to share some of my own opinions which may not
differ alot from you people...just more concise, and may be
easier to explain to a primary school student. Not guaranteed
though!
Imagine you have 1 piece of something. Now, for the sake of
argument, you are to divide by 1/2 piece. Which means to ask how
many 1/2 pieces are there in 1 piece. Since 1/2 piece is 1 piece
split into 2 equal and smaller pieces, there 2 of the 1/2 pieces
present in 1 piece. One can use this argument to
generalise and get:
1 (divide) 1/n =
n
(1.1)
Then, you can multiply by a factor of m, which means to say to
add 1 by m times. Since in 1 piece there are n number of 1/n, in m
pieces there are mn number of 1/n pieces and so,
m (divide) 1/n =
mn
(1.2)
Clearly, the statement is numerically
equvilent to saying having 1 piece divided by 1/mn. Which can be
easily seen from equation (1.1). Intuitively, since n is added
m times, the number of pieces to split up is increased by m
times.
1 (divide) 1/mn =
mn
(1.3)
Next if 1 piece is split into p/n, which means 1/n added p
times, then
1 (divide) p/n =
n/p
(1.4)
Intuitively, when you count n pieces of 1/n you get 1
piece, and p/n is already the value when you count p times of
1/n. So now when you count n times, you get p pieces instead.
So to get 1 piece, you have to divide by p times now.
Putting these ideas together and you see that the general rule
of thumb is that you have to overturn the fraction after the
division sign! 
You want to explain all this to a primary school kid? 
